We study f-vectors, which are the maximal degree vectors of F-polynomials in cluster algebra theory. For a cluster algebra of finite type, we find that positive f-vectors correspond with d-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of d-vectors, we prove that cluster variables in a cluster are uniquely determined by their f-vectors when the cluster algebra is of finite type or rank 2.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Buan, A. B., Iyama, O., Reiten, I., Scott, J., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., 145, (2009), no. 4, 1035–1079, https://doi.org/10.1112/S0010437X09003960
Buan, A. B., Marsh, R., Reineke, M., Reiten, I., Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618
Cao, P., Li, F., The enough \(g\)-pairs property and denominator vectors of cluster algebras, Trans. Amer. Math. Soc. 377 (2020), 1547–1572
Ceballos, C., Pilaud, V., Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1421–1439
Çanakçı, Í., Schiffler, R., Snake graphs and continued fractions, 2017. preprint, arXiv:1711.02461
Fock, V. V., Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ecole Normale. Sup. 42 (2009), no. 6, 865–930
Fu, C., Geng, S., On indecomposable \(\tau \)-rigid modules over cluster-tilted algebras of tame type, 2017. preprint, arXiv:1705.10939
Fujiwara, S., Gyoda, Y., Duality between final-seed and initial-seed mutations in cluster algebras, SIGMA, 15 (2019), 24 pages
Fu, C., Geng, S., Liu, P., Cluster algebras arising from cluster tubes I: integer vectors, 2018. preprint, arXiv:1801.00709
Fu, C., Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), no. 2, 859–895
Fomin, S., Shapiro, M., Thurston, D., Cluster algebras and triangulated surfaces. part I: Cluster complexes, Acta Math. 201 (2008), 83–146
Fomin, S., Zelevinsky, A., Cluster Algebra I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529
Fomin, S., Zelevinsky, A., Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63–121
Fomin, S., Zelevinsky, A., Cluster Algebra IV: Coefficients, Comp. Math. 143 (2007), 112–164
Geng, S., Peng, L., The dimension vectors of indecomposable modules of cluster-tilted algebras and the Fomin-Zelevinsky denominators conjecture, Acta. Math. 28 (2012), no. 3, 581–586
Gyoda, Y., Yurikusa, T., \(F\)-matrices of cluster algebras from triangulated surfaces, Ann. Comb. 24 (2020), no. 4, 649–695
Inoue, R., Iyama, O., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T-systems and Y-systems, Nagoya Math. J. 197 (2010), 59–174
Inoue, R., Iyama, O., Keller, B., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type \(B_r\), Publ. RIMS. 49 (2013), 1–42
Inoue, R., Iyama, O., Keller, B., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Type \(C_r,F_4,G_2\), Publ. RIMS. 49 (2013), 43–85
Lee, K., Li, L., Zelevinsky, A., Greedy elements in rank 2 cluster algebras, Selecta Mathematica. 20 (2014), no. 1, 57–82
Lee, K., Schiffler, R., A combinatorial formula for rank \(2\) cluster variables, J. Algebraic Combin. 37 (2013), no. 1, 67–85
Nakanishi, T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J. 202 (2011), 23–43
Nakanishi, T., Stella, S., Diagrammatic description of \(c\)-vectors and \(d\)-vectors of cluster algebras of finite type, Electron. J. Comb. 21 (2014), no. 2, 107 pages
Ringel, C. M., Cluster-concealed algebras, Adv. Math. 226 (2011), no. 2, 1513–1537
Rabideau, M., Schiffler, R., Continued fractions and orderings on the Markov numbers, 2018. preprint, arXiv:1801.07155
Reading, N., Stella, S., Initial-seed recursions and dualities for d-vectors, Pacific J. Math. 293 (2018), 179–206
Sherman, P., Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J. 4 (2004), no. 4, 947–974
The author would like to express his gratitude to Bernhard Keller for insightful comments about Theorem 1.8. The author appreciates important remarks about Conjecture 1.10 by Changjian Fu. Toshiya Yurikusa gives helpful advice about Theorem 1.11. The author received generous support from Tomoki Nakanishi. The author also thanks Haruhisa Enomoto, Yoshiki Aibara, and Naohiro Tsuzu. This work was supported by JSPS KAKENHI Grant number JP20J12675.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Alexander Yong
About this article
Cite this article
Gyoda, Y. Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2. Ann. Comb. 25, 573–594 (2021). https://doi.org/10.1007/s00026-021-00527-6
Mathematics Subject Classification
- Cluster algebra